It is fairly simple - here are the steps
On current phone
Settings - Notes - on my phone account - toggle to OFF - this ensures that you write notes only to the cloud backed up area
Settings - iCloud - Notes - make sure it is toggled ON for back up
Settings - iCloud - Backup - Turn it ON - use backup now to force a manual backup
Preferably connect your phone to Power and be in a wifi zone
Once backup completes - log into https://www.icloud.com with your Apple ID credentials and see if your notes are there
If yes - then syncing the new phone will restore them